Applicable Algebra in Engineering Communication and Computing最新文献

The projective norm graphs are central objects to extremal combinatorics. They appear in a variety of contexts, most importantly they provide tight constructions for the Turán number of complete bipartite graphs (K_{t,s}) with (s>(t-1)!). In this note we deepen their understanding further by determining their automorphism group.

In 2016, Y. Kumar et al. in the paper ‘Affine equivalence and non-linearity of permutations over ({mathbb {Z}}_n)’ conjectured that: For (nge 3), the nonlinearity of any permutation on ({mathbb {Z}}_n), the ring of integers modulo n, cannot exceed (n-2). For an odd prime p, we settle the above conjecture when (n=2p) and for (pequiv 3 pmod {4}) we prove the above conjecture with an improved upper bound. Further, we derive a lower bound on (max {mathcal {N}}{mathcal {L}}_n) when n is an odd prime or twice of an odd prime where (max {mathcal {N}}{mathcal {L}}_n) denotes the maximum possible nonlinearity of any permutation on ({mathbb {Z}}_n).

In this work, we give three new techniques for constructing Hermitian self-dual codes over commutative Frobenius rings with a non-trivial involutory automorphism using (lambda)-circulant matrices. The new constructions are derived as modifications of various well-known circulant constructions of self-dual codes. Applying these constructions together with the building-up construction, we construct many new best known quaternary Hermitian self-dual codes of lengths 26, 32, 36, 38 and 40.

Let p be a prime and n be a positive integer. We consider rational functions (f_b(X)=X+1/(X^p-X+b)) over ({mathbb {F}}_{p^n}) with (textrm{Tr}(b)ne 0). In Hou and Sze (Finite Fields Appl 68, Paper No. 10175, 2020), it is shown that (f_b(X)) is not a permutation for (p>3) and (nge 5), while it is for (p=2,3) and (nge 1). It is conjectured that (f_b(X)) is also not a permutation for (p>3) and (n=3,4), which was recently proved sufficiently large primes in Bartoli and Hou (Finite Fields Appl 76, Paper No. 101904, 2021). In this note, we give a new proof for the fact that (f_b(X)) is not a permutation for (p>3) and (nge 5). With this proof, we also show the existence of many elements (bin {mathbb {F}}_{p^n}) for which (f_b(X)) is not a permutation for (n=3,4).

An ideal ({{mathcal {I}}}) of ({mathbb {Q}}[x_1,dots ,x_n]) is said to be m-absorbing if any monomial of total degree (p>m) belongs to ({{mathcal {I}}}) and if there is a monomial M of total degree m such that (Mnot in {{mathcal {I}}}.) Inspired by the fundamental work of Lechuga and Murillo (Topology 39:89–94, 2000) who established a connection between graph theory and rational homotopy theory, these paper aims to characterise the m-absorbing ideals of ({mathbb {Q}}[x_1,dots ,x_n]) generated by a family of complete homogeneous symmetric polynomials of the form (P^{(k)}_{rs}=sum _{t=1}^{k}x^{k-t}_{r}x^{t-1}_{s}), where (kge 3.)

We revisit the seminal Brill–Noether algorithm for plane curves with ordinary singularities. Our new approach takes advantage of fast algorithms for polynomials and structured matrices. We design a new probabilistic algorithm of Las Vegas type that computes a Riemann–Roch space in expected sub-quadratic time.

Moved by a question posed us by Wolfgang Rump, we investigate the Rump ideal ({mathbb {I}}(p^2-pq+qp)subset {mathbb {Z}}langle q,q^{-1}, prangle ) and we show, this way, the power of Zacharias representation.

Computer Algebra relies heavily on the computation of Gröbner bases, and these computations are primarily performed by means of Buchberger’s algorithm. In this overview paper, we focus on methods avoiding the computational intensity associated to Buchberger’s algorithm and, in most cases, even avoiding the concept of Gröbner bases, in favour of methods relying on linear algebra and combinatorics.

Let r be a divisor of (q-1.) An element (alpha in {mathbb {F}}_{q}) is said to be r-primitive if ord((alpha )=frac{q-1}{r}). In this paper, we discuss the existence of r-primitive pairs ((alpha , f(alpha ))) where (alpha in {mathbb {F}}_q), f(x) is a general rational function of degree sum m (degree sum is the sum of the degrees of numerator and denominator of f(x)) and the denominator of f(x) is square-free. Then we show that for any integer (m>0), there exists a positive constant (B_{r,m}) such that if (q>B_{r,m}), then such r-primitive pairs exist. In particular, we present a bound for (B_{r,m}) with (r=2) and (min {2,3,4,5,6}), and provide some conditions on the existence of 2-primitive pairs.